Problem: Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{k^2 + 8k}{k^2 + 13k + 40}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{k^2 + 8k}{k^2 + 13k + 40} = \dfrac{(k)(k + 8)}{(k + 5)(k + 8)} $ Notice that the term $(k + 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(k + 8)$ gives: $x = \dfrac{k}{k + 5}$ Since we divided by $(k + 8)$, $k \neq -8$. $x = \dfrac{k}{k + 5}; \space k \neq -8$